Abstract

A wavelet element method is developed for analyzing lamellar diffraction gratings or grating stacks. The eigenmodes of the grating layers are accurately calculated by this method, and then the diffraction efficiencies of the gratings are calculated by the S-matrix algorithm. The method proposed in this paper consists in mapping each homogeneous layer to a wavelet element, and then matching them according to the boundary conditions between the layers. By this method the boundary conditions are satisfied rigorously and the Gibbs phenomenon in the Fourier modal method (FMM) can be avoided. The method performs better than the standard FMM for gratings involving metals. It can also be applied to analyze other discontinuous structures.

© 2013 Optical Society of America

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References

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  1. K. Knop, “Rigorous diffraction theory for transmission phase gratings with deep rectangular grooves,” J. Opt. Soc. Am. 68, 1206–1210 (1978).
    [CrossRef]
  2. G. Granet and B. Guizal, “Efficient implementation of the coupled-wave method for metallic lamellar gratings in TM polarization,” J. Opt. Soc. Am. A 13, 1019–1023 (1996).
    [CrossRef]
  3. P. Lalanne and G. M. Morris, “Highly improved convergence of the coupled-wave method for TM polarization,” J. Opt. Soc. Am. A 13, 779–784 (1996).
    [CrossRef]
  4. L. Li, “Use of Fourier series in the analysis of discontinuous periodic structures,” J. Opt. Soc. Am. A 13, 1870–1876 (1996).
    [CrossRef]
  5. E. Popov, M. Nevière, and N. Bonod, “Factorization of products of discontinuous functions applied to Fourier-Bessel basis,” J. Opt. Soc. Am. A 21, 46–52 (2004).
    [CrossRef]
  6. K. Edee, “Modal method based on subsectional Gegenbauer polynomial expansion for lamellar gratings,” J. Opt. Soc. Am. A 28, 2006–2013 (2011).
    [CrossRef]
  7. J. W. Gibbs, “Fourier’s series,” Nature 59, 200 (1898).
    [CrossRef]
  8. J. W. Gibbs, “Fourier’s series,” Nature 59, 606 (1899).
    [CrossRef]
  9. X. Checoury and J.-M. Lourtioz, “Wavelet method for computing band diagrams of 2D photonic crystals,” Opt. Commun. 259, 360–365 (2006).
    [CrossRef]
  10. I. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, “The dielectric lamellar diffraction grating,” Optica Acta 28, 413–428 (1981).
    [CrossRef]
  11. R. H. Morf, “Exponentially convergent and numerically efficient solution of Maxwell’s equations for lamellar gratings,” J. Opt. Soc. Am. A 12, 1043–1056 (1995).
    [CrossRef]
  12. D. Song and Y. Y. Lu, “High-order finite difference modal method for diffraction gratings,” J. Mod. Opt. 59, 800–808 (2012).
    [CrossRef]
  13. P. Lalanne and J.-P. Hugonin, “Numerical performance of finite-difference modal methods for the electromagnetic analysis of one-dimensional lamellar gratings,” J. Opt. Soc. Am. A 17, 1033–1042 (2000).
    [CrossRef]
  14. D. Song, L. Yuan, and Y. Y. Lu, “Fourier-matching pseudospectral modal method for diffraction gratings,” J. Opt. Soc. Am. A 28, 613–620 (2011).
    [CrossRef]
  15. G. Granet, “Fourier-matching pseudospectral modal method for diffraction gratings: comment,” J. Opt. Soc. Am. A 29, 1843–1845 (2012).
    [CrossRef]
  16. A. M. Armeanu, K. Edee, G. Grnaet, and P. Schiavone, “Modal method based on spline expansion for the electronagnetic analysis of the lamellar grating,” Progress Electromagn. Res. 106, 243–261 (2010).
    [CrossRef]
  17. K. Edee, I. Fenniche, G. Granet, and B. Guizal, “Modal method based on subsectional Gegenbauer polynomial expansion for lamellar gratings: weighting function, convergence and stability,” Progress Electromagn. Res. 133, 17–35 (2013).
    [CrossRef]
  18. J. Yuan and Y. Y. Lu, “Photonic bandgap calculations with Dirichlet-to-Neumann maps,” J. Opt. Soc. Am. A 23, 3217–3222 (2006).
    [CrossRef]
  19. W. Dahmen, A. Kunoth, and K. Urban, “Biorthogonal spline wavelets on the interval—stability and moment conditions,” Appl. Comput. Harmon. Anal. 6, 132–196 (1999).
    [CrossRef]
  20. C. Canuto, A. Tabacco, and K. Urban, “The wavelet element method: part I. construction and analysis,” Appl. Comput. Harmon. Anal. 6, 1–52 (1999).
    [CrossRef]
  21. R. F. Harrington, Field Computation by Moment Methods(Wiley-IEEE, 1993), p. 229.
  22. K. Urban, Wavelet Methods for Elliptic Partial Differential Equations (Oxford University, 2009).
  23. A. Cohen, I. Daubechies, and J. C. Feauveau, “Biorthogonal bases of compactly supported wavelets,” Commun. Pure Appl. Math. 45, 485–560 (1992).
    [CrossRef]
  24. S. Bertoluzza, C. Canuto, and K. Urban, “On the adaptive computation of integrals of wavelets,” Appl. Numer. Math. 34, 13–38 (2000).
    [CrossRef]
  25. W. Dahmen and C. A. Micchelli, “Using the refinement equation for evaluating integrals of wavelets,” SIAM J. Numer. Anal. 30, 507–537 (1993).
    [CrossRef]
  26. L. Li, “Formulation and comparison of two recursive matrix algorithms for modeling layered diffraction gratings,” J. Opt. Soc. Am. A 13, 1024–1035 (1996).
    [CrossRef]
  27. K. Edee, J. P. Plumey, and J. Chandezon, “On the Rayleigh–Fourier method and the Chandezon method: comparative study,” Opt. Commun. 286, 34–41 (2013).
    [CrossRef]
  28. S. T. Peng, T. Tamir, and H. L. Bertoni, “Theory of periodic dielect waveguides,” IEEE Trans. Microwave Theor. Tech. 23, 123–133 (1975).
    [CrossRef]
  29. G. Granet, “Reformulation of the lamellar grating problem through the concept of adaptive spatial resolution,” J. Opt. Soc. Am. A 16, 2510–2516 (1999).
    [CrossRef]

2013 (2)

K. Edee, I. Fenniche, G. Granet, and B. Guizal, “Modal method based on subsectional Gegenbauer polynomial expansion for lamellar gratings: weighting function, convergence and stability,” Progress Electromagn. Res. 133, 17–35 (2013).
[CrossRef]

K. Edee, J. P. Plumey, and J. Chandezon, “On the Rayleigh–Fourier method and the Chandezon method: comparative study,” Opt. Commun. 286, 34–41 (2013).
[CrossRef]

2012 (2)

G. Granet, “Fourier-matching pseudospectral modal method for diffraction gratings: comment,” J. Opt. Soc. Am. A 29, 1843–1845 (2012).
[CrossRef]

D. Song and Y. Y. Lu, “High-order finite difference modal method for diffraction gratings,” J. Mod. Opt. 59, 800–808 (2012).
[CrossRef]

2011 (2)

2010 (1)

A. M. Armeanu, K. Edee, G. Grnaet, and P. Schiavone, “Modal method based on spline expansion for the electronagnetic analysis of the lamellar grating,” Progress Electromagn. Res. 106, 243–261 (2010).
[CrossRef]

2006 (2)

X. Checoury and J.-M. Lourtioz, “Wavelet method for computing band diagrams of 2D photonic crystals,” Opt. Commun. 259, 360–365 (2006).
[CrossRef]

J. Yuan and Y. Y. Lu, “Photonic bandgap calculations with Dirichlet-to-Neumann maps,” J. Opt. Soc. Am. A 23, 3217–3222 (2006).
[CrossRef]

2004 (1)

2000 (2)

1999 (3)

G. Granet, “Reformulation of the lamellar grating problem through the concept of adaptive spatial resolution,” J. Opt. Soc. Am. A 16, 2510–2516 (1999).
[CrossRef]

W. Dahmen, A. Kunoth, and K. Urban, “Biorthogonal spline wavelets on the interval—stability and moment conditions,” Appl. Comput. Harmon. Anal. 6, 132–196 (1999).
[CrossRef]

C. Canuto, A. Tabacco, and K. Urban, “The wavelet element method: part I. construction and analysis,” Appl. Comput. Harmon. Anal. 6, 1–52 (1999).
[CrossRef]

1996 (4)

1995 (1)

1993 (1)

W. Dahmen and C. A. Micchelli, “Using the refinement equation for evaluating integrals of wavelets,” SIAM J. Numer. Anal. 30, 507–537 (1993).
[CrossRef]

1992 (1)

A. Cohen, I. Daubechies, and J. C. Feauveau, “Biorthogonal bases of compactly supported wavelets,” Commun. Pure Appl. Math. 45, 485–560 (1992).
[CrossRef]

1981 (1)

I. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, “The dielectric lamellar diffraction grating,” Optica Acta 28, 413–428 (1981).
[CrossRef]

1978 (1)

1975 (1)

S. T. Peng, T. Tamir, and H. L. Bertoni, “Theory of periodic dielect waveguides,” IEEE Trans. Microwave Theor. Tech. 23, 123–133 (1975).
[CrossRef]

1899 (1)

J. W. Gibbs, “Fourier’s series,” Nature 59, 606 (1899).
[CrossRef]

1898 (1)

J. W. Gibbs, “Fourier’s series,” Nature 59, 200 (1898).
[CrossRef]

Adams, J. L.

I. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, “The dielectric lamellar diffraction grating,” Optica Acta 28, 413–428 (1981).
[CrossRef]

Andrewartha, J. R.

I. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, “The dielectric lamellar diffraction grating,” Optica Acta 28, 413–428 (1981).
[CrossRef]

Armeanu, A. M.

A. M. Armeanu, K. Edee, G. Grnaet, and P. Schiavone, “Modal method based on spline expansion for the electronagnetic analysis of the lamellar grating,” Progress Electromagn. Res. 106, 243–261 (2010).
[CrossRef]

Bertoluzza, S.

S. Bertoluzza, C. Canuto, and K. Urban, “On the adaptive computation of integrals of wavelets,” Appl. Numer. Math. 34, 13–38 (2000).
[CrossRef]

Bertoni, H. L.

S. T. Peng, T. Tamir, and H. L. Bertoni, “Theory of periodic dielect waveguides,” IEEE Trans. Microwave Theor. Tech. 23, 123–133 (1975).
[CrossRef]

Bonod, N.

Botten, I. C.

I. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, “The dielectric lamellar diffraction grating,” Optica Acta 28, 413–428 (1981).
[CrossRef]

Canuto, C.

S. Bertoluzza, C. Canuto, and K. Urban, “On the adaptive computation of integrals of wavelets,” Appl. Numer. Math. 34, 13–38 (2000).
[CrossRef]

C. Canuto, A. Tabacco, and K. Urban, “The wavelet element method: part I. construction and analysis,” Appl. Comput. Harmon. Anal. 6, 1–52 (1999).
[CrossRef]

Chandezon, J.

K. Edee, J. P. Plumey, and J. Chandezon, “On the Rayleigh–Fourier method and the Chandezon method: comparative study,” Opt. Commun. 286, 34–41 (2013).
[CrossRef]

Checoury, X.

X. Checoury and J.-M. Lourtioz, “Wavelet method for computing band diagrams of 2D photonic crystals,” Opt. Commun. 259, 360–365 (2006).
[CrossRef]

Cohen, A.

A. Cohen, I. Daubechies, and J. C. Feauveau, “Biorthogonal bases of compactly supported wavelets,” Commun. Pure Appl. Math. 45, 485–560 (1992).
[CrossRef]

Craig, M. S.

I. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, “The dielectric lamellar diffraction grating,” Optica Acta 28, 413–428 (1981).
[CrossRef]

Dahmen, W.

W. Dahmen, A. Kunoth, and K. Urban, “Biorthogonal spline wavelets on the interval—stability and moment conditions,” Appl. Comput. Harmon. Anal. 6, 132–196 (1999).
[CrossRef]

W. Dahmen and C. A. Micchelli, “Using the refinement equation for evaluating integrals of wavelets,” SIAM J. Numer. Anal. 30, 507–537 (1993).
[CrossRef]

Daubechies, I.

A. Cohen, I. Daubechies, and J. C. Feauveau, “Biorthogonal bases of compactly supported wavelets,” Commun. Pure Appl. Math. 45, 485–560 (1992).
[CrossRef]

Edee, K.

K. Edee, J. P. Plumey, and J. Chandezon, “On the Rayleigh–Fourier method and the Chandezon method: comparative study,” Opt. Commun. 286, 34–41 (2013).
[CrossRef]

K. Edee, I. Fenniche, G. Granet, and B. Guizal, “Modal method based on subsectional Gegenbauer polynomial expansion for lamellar gratings: weighting function, convergence and stability,” Progress Electromagn. Res. 133, 17–35 (2013).
[CrossRef]

K. Edee, “Modal method based on subsectional Gegenbauer polynomial expansion for lamellar gratings,” J. Opt. Soc. Am. A 28, 2006–2013 (2011).
[CrossRef]

A. M. Armeanu, K. Edee, G. Grnaet, and P. Schiavone, “Modal method based on spline expansion for the electronagnetic analysis of the lamellar grating,” Progress Electromagn. Res. 106, 243–261 (2010).
[CrossRef]

Feauveau, J. C.

A. Cohen, I. Daubechies, and J. C. Feauveau, “Biorthogonal bases of compactly supported wavelets,” Commun. Pure Appl. Math. 45, 485–560 (1992).
[CrossRef]

Fenniche, I.

K. Edee, I. Fenniche, G. Granet, and B. Guizal, “Modal method based on subsectional Gegenbauer polynomial expansion for lamellar gratings: weighting function, convergence and stability,” Progress Electromagn. Res. 133, 17–35 (2013).
[CrossRef]

Gibbs, J. W.

J. W. Gibbs, “Fourier’s series,” Nature 59, 606 (1899).
[CrossRef]

J. W. Gibbs, “Fourier’s series,” Nature 59, 200 (1898).
[CrossRef]

Granet, G.

Grnaet, G.

A. M. Armeanu, K. Edee, G. Grnaet, and P. Schiavone, “Modal method based on spline expansion for the electronagnetic analysis of the lamellar grating,” Progress Electromagn. Res. 106, 243–261 (2010).
[CrossRef]

Guizal, B.

K. Edee, I. Fenniche, G. Granet, and B. Guizal, “Modal method based on subsectional Gegenbauer polynomial expansion for lamellar gratings: weighting function, convergence and stability,” Progress Electromagn. Res. 133, 17–35 (2013).
[CrossRef]

G. Granet and B. Guizal, “Efficient implementation of the coupled-wave method for metallic lamellar gratings in TM polarization,” J. Opt. Soc. Am. A 13, 1019–1023 (1996).
[CrossRef]

Harrington, R. F.

R. F. Harrington, Field Computation by Moment Methods(Wiley-IEEE, 1993), p. 229.

Hugonin, J.-P.

Knop, K.

Kunoth, A.

W. Dahmen, A. Kunoth, and K. Urban, “Biorthogonal spline wavelets on the interval—stability and moment conditions,” Appl. Comput. Harmon. Anal. 6, 132–196 (1999).
[CrossRef]

Lalanne, P.

Li, L.

Lourtioz, J.-M.

X. Checoury and J.-M. Lourtioz, “Wavelet method for computing band diagrams of 2D photonic crystals,” Opt. Commun. 259, 360–365 (2006).
[CrossRef]

Lu, Y. Y.

McPhedran, R. C.

I. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, “The dielectric lamellar diffraction grating,” Optica Acta 28, 413–428 (1981).
[CrossRef]

Micchelli, C. A.

W. Dahmen and C. A. Micchelli, “Using the refinement equation for evaluating integrals of wavelets,” SIAM J. Numer. Anal. 30, 507–537 (1993).
[CrossRef]

Morf, R. H.

Morris, G. M.

Nevière, M.

Peng, S. T.

S. T. Peng, T. Tamir, and H. L. Bertoni, “Theory of periodic dielect waveguides,” IEEE Trans. Microwave Theor. Tech. 23, 123–133 (1975).
[CrossRef]

Plumey, J. P.

K. Edee, J. P. Plumey, and J. Chandezon, “On the Rayleigh–Fourier method and the Chandezon method: comparative study,” Opt. Commun. 286, 34–41 (2013).
[CrossRef]

Popov, E.

Schiavone, P.

A. M. Armeanu, K. Edee, G. Grnaet, and P. Schiavone, “Modal method based on spline expansion for the electronagnetic analysis of the lamellar grating,” Progress Electromagn. Res. 106, 243–261 (2010).
[CrossRef]

Song, D.

D. Song and Y. Y. Lu, “High-order finite difference modal method for diffraction gratings,” J. Mod. Opt. 59, 800–808 (2012).
[CrossRef]

D. Song, L. Yuan, and Y. Y. Lu, “Fourier-matching pseudospectral modal method for diffraction gratings,” J. Opt. Soc. Am. A 28, 613–620 (2011).
[CrossRef]

Tabacco, A.

C. Canuto, A. Tabacco, and K. Urban, “The wavelet element method: part I. construction and analysis,” Appl. Comput. Harmon. Anal. 6, 1–52 (1999).
[CrossRef]

Tamir, T.

S. T. Peng, T. Tamir, and H. L. Bertoni, “Theory of periodic dielect waveguides,” IEEE Trans. Microwave Theor. Tech. 23, 123–133 (1975).
[CrossRef]

Urban, K.

S. Bertoluzza, C. Canuto, and K. Urban, “On the adaptive computation of integrals of wavelets,” Appl. Numer. Math. 34, 13–38 (2000).
[CrossRef]

C. Canuto, A. Tabacco, and K. Urban, “The wavelet element method: part I. construction and analysis,” Appl. Comput. Harmon. Anal. 6, 1–52 (1999).
[CrossRef]

W. Dahmen, A. Kunoth, and K. Urban, “Biorthogonal spline wavelets on the interval—stability and moment conditions,” Appl. Comput. Harmon. Anal. 6, 132–196 (1999).
[CrossRef]

K. Urban, Wavelet Methods for Elliptic Partial Differential Equations (Oxford University, 2009).

Yuan, J.

Yuan, L.

Appl. Comput. Harmon. Anal. (2)

W. Dahmen, A. Kunoth, and K. Urban, “Biorthogonal spline wavelets on the interval—stability and moment conditions,” Appl. Comput. Harmon. Anal. 6, 132–196 (1999).
[CrossRef]

C. Canuto, A. Tabacco, and K. Urban, “The wavelet element method: part I. construction and analysis,” Appl. Comput. Harmon. Anal. 6, 1–52 (1999).
[CrossRef]

Appl. Numer. Math. (1)

S. Bertoluzza, C. Canuto, and K. Urban, “On the adaptive computation of integrals of wavelets,” Appl. Numer. Math. 34, 13–38 (2000).
[CrossRef]

Commun. Pure Appl. Math. (1)

A. Cohen, I. Daubechies, and J. C. Feauveau, “Biorthogonal bases of compactly supported wavelets,” Commun. Pure Appl. Math. 45, 485–560 (1992).
[CrossRef]

IEEE Trans. Microwave Theor. Tech. (1)

S. T. Peng, T. Tamir, and H. L. Bertoni, “Theory of periodic dielect waveguides,” IEEE Trans. Microwave Theor. Tech. 23, 123–133 (1975).
[CrossRef]

J. Mod. Opt. (1)

D. Song and Y. Y. Lu, “High-order finite difference modal method for diffraction gratings,” J. Mod. Opt. 59, 800–808 (2012).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (12)

G. Granet and B. Guizal, “Efficient implementation of the coupled-wave method for metallic lamellar gratings in TM polarization,” J. Opt. Soc. Am. A 13, 1019–1023 (1996).
[CrossRef]

P. Lalanne and G. M. Morris, “Highly improved convergence of the coupled-wave method for TM polarization,” J. Opt. Soc. Am. A 13, 779–784 (1996).
[CrossRef]

L. Li, “Use of Fourier series in the analysis of discontinuous periodic structures,” J. Opt. Soc. Am. A 13, 1870–1876 (1996).
[CrossRef]

E. Popov, M. Nevière, and N. Bonod, “Factorization of products of discontinuous functions applied to Fourier-Bessel basis,” J. Opt. Soc. Am. A 21, 46–52 (2004).
[CrossRef]

K. Edee, “Modal method based on subsectional Gegenbauer polynomial expansion for lamellar gratings,” J. Opt. Soc. Am. A 28, 2006–2013 (2011).
[CrossRef]

P. Lalanne and J.-P. Hugonin, “Numerical performance of finite-difference modal methods for the electromagnetic analysis of one-dimensional lamellar gratings,” J. Opt. Soc. Am. A 17, 1033–1042 (2000).
[CrossRef]

D. Song, L. Yuan, and Y. Y. Lu, “Fourier-matching pseudospectral modal method for diffraction gratings,” J. Opt. Soc. Am. A 28, 613–620 (2011).
[CrossRef]

G. Granet, “Fourier-matching pseudospectral modal method for diffraction gratings: comment,” J. Opt. Soc. Am. A 29, 1843–1845 (2012).
[CrossRef]

G. Granet, “Reformulation of the lamellar grating problem through the concept of adaptive spatial resolution,” J. Opt. Soc. Am. A 16, 2510–2516 (1999).
[CrossRef]

L. Li, “Formulation and comparison of two recursive matrix algorithms for modeling layered diffraction gratings,” J. Opt. Soc. Am. A 13, 1024–1035 (1996).
[CrossRef]

R. H. Morf, “Exponentially convergent and numerically efficient solution of Maxwell’s equations for lamellar gratings,” J. Opt. Soc. Am. A 12, 1043–1056 (1995).
[CrossRef]

J. Yuan and Y. Y. Lu, “Photonic bandgap calculations with Dirichlet-to-Neumann maps,” J. Opt. Soc. Am. A 23, 3217–3222 (2006).
[CrossRef]

Nature (2)

J. W. Gibbs, “Fourier’s series,” Nature 59, 200 (1898).
[CrossRef]

J. W. Gibbs, “Fourier’s series,” Nature 59, 606 (1899).
[CrossRef]

Opt. Commun. (2)

X. Checoury and J.-M. Lourtioz, “Wavelet method for computing band diagrams of 2D photonic crystals,” Opt. Commun. 259, 360–365 (2006).
[CrossRef]

K. Edee, J. P. Plumey, and J. Chandezon, “On the Rayleigh–Fourier method and the Chandezon method: comparative study,” Opt. Commun. 286, 34–41 (2013).
[CrossRef]

Optica Acta (1)

I. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, “The dielectric lamellar diffraction grating,” Optica Acta 28, 413–428 (1981).
[CrossRef]

Progress Electromagn. Res. (2)

A. M. Armeanu, K. Edee, G. Grnaet, and P. Schiavone, “Modal method based on spline expansion for the electronagnetic analysis of the lamellar grating,” Progress Electromagn. Res. 106, 243–261 (2010).
[CrossRef]

K. Edee, I. Fenniche, G. Granet, and B. Guizal, “Modal method based on subsectional Gegenbauer polynomial expansion for lamellar gratings: weighting function, convergence and stability,” Progress Electromagn. Res. 133, 17–35 (2013).
[CrossRef]

SIAM J. Numer. Anal. (1)

W. Dahmen and C. A. Micchelli, “Using the refinement equation for evaluating integrals of wavelets,” SIAM J. Numer. Anal. 30, 507–537 (1993).
[CrossRef]

Other (2)

R. F. Harrington, Field Computation by Moment Methods(Wiley-IEEE, 1993), p. 229.

K. Urban, Wavelet Methods for Elliptic Partial Differential Equations (Oxford University, 2009).

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Figures (5)

Fig. 1.
Fig. 1.

Grating configuration.

Fig. 2.
Fig. 2.

Some scaling and wavelet functions on level 5.

Fig. 3.
Fig. 3.

Convergence of the eigenvalue with the smallest modulus for (a) TE polarization and (b) TM polarization. The exact eigenvalue is found to be 6.495282130093448+0.182874367618269i (TE) and 7.867429502364694+0.272265173640988i (TM).

Fig. 4.
Fig. 4.

First-order derivative of the eigenfunction g1(x) with the eigenvalue [β1(1)]2=7.8674+0.2723i.

Fig. 5.
Fig. 5.

Example 3 (TM case): diffraction efficiency of the zeroth reflected order calculated by the FMM (circles) and the WEM (bior5.5) (crosses).

Tables (7)

Tables Icon

Table 1. Example 1 (TM case): Diffraction Efficiency of the Zeroth Reflected Order (RE0) Calculated by the FMM, the WEM (bior3.5), and the WEM (bior5.5)

Tables Icon

Table 2. Example 1 (TE case): Diffraction Efficiency of the Minus-First Reflected Order (RE1) Calculated by the FMM, the WEM (bior3.5), and the WEM (bior5.5)

Tables Icon

Table 3. Example 2 (TM case): Diffraction Efficiency of the Zeroth Reflected Order (RE0) Calculated by the FMM, the WEM (bior3.5), and the WEM (bior5.5)

Tables Icon

Table 4. Example 2 (TE case): Diffraction Efficiency of the Minus-First Reflected Order (RE1) Calculated by the FMM, the WEM (bior3.5), and the WEM (bior5.5)

Tables Icon

Table 5. Example 3 (TM case): Diffraction Efficiency of the Zeroth Reflected Order (RE1) Calculated by the FMM, the WEM (bior3.5), and the WEM (bior5.5)

Tables Icon

Table 6. Example 3 (TE case): Diffraction Efficiency of the Minus-First Reflected Order (RE1) Calculated by the FMM, the WEM (bior3.5), and the WEM (bior5.5)

Tables Icon

Table 7. Matrix Size of Different Methods for Four Correct Digits in Examples 1 and 3

Equations (47)

Equations on this page are rendered with MathJax. Learn more.

υ1(x)={υ11,x[0,pL]υ12,x(pL,L],
TM:x[1υ12ux]+y[1υ12uy]+k02u=0,
TE:2ux2+2uy2+k02υ12u=0,
u(i)=exp{i[α0xβ0(2)(yh)]}inD2.
α0=k0υ2sinθ,β0(2)=k0υ2cosθ.
u(r)=m=Rmexp{i[αmx+βm(2)(yh)]},inD2,
u(t)=m=Tmexp[iαmxβm(0)y],inD0,
αm=α0+2πmL,βm(2)={k02υ22αm2,for|αm|<k0υ2,iαm2k02υ22,for|αm|>k0υ2,βm(0)={k02υ02αm2,for|αm|<k0υ0,iαm2k02υ02,for|αm|>k0υ0,
u=l=1{alexp(iβl(1)y)+blexp[iβl(1)(hy)]}gl(x),
υ12ddx(1υ12dgdx)+k02υ12g=β2g,
g(pL)=g(pL+),1υ12(pL)dgdx(pL)=1υ12(pL+)dgdx(pL+),
eiα0Lg(0+)=g(L),eiα0Lυ12(0+)dgdx(0+)=1υ12(L)dgdx(L),
u(2)(x,h+)=u(1)(x,h),1υ22u(2)y(x,h+)=1υ12(x,h)u(1)y(x,h),
u(1)(x,0+)=u(0)(x,0),1υ12(x,0)u(1)y(x,0+)=1υ02u(0)y(x,0),
Ψj0,J:=Φj0j=j0JΨj,Ψ˜j0,J:=Φ˜j0j=j0JΨ˜j,
fi(0)0orfi(0)0i=1,2,fi(1)0orfi(1)0i=m1,m.
gl(x)=mcmlbm,
mcml[υ12ddx(1υ12dbmdx)+k02υ12bm]=β2mcmlbm.
mcml(υ12ddx(1υ12dbmdx)+k02υ12bm,b˜n)=β2mcml(bm,b˜n).
mcml(dbmdxb˜n|Γ(dbmdx,db˜ndx)+k02(υ12bm,b˜n))=β2mcml(bm,b˜n),
dbmdxb˜n|Γ=Anm(1),
(dbmdx,db˜ndx)=Anm(2),
(υ12bm,b˜n)=Anm(3).
[A(1)+A(2)+k02A(3)]cl=β2cl,
φ(x)=2nZhnφ(2xn),ψ(x)=2nZgnφ(2xn),
F(z1,z2)=01φ(xz1)φ˜(xz2)dx,ziR.
u=l=1M{alexp(iβl(1)y)+blexp[iβl(1)(hy)]}gl(x).
[II1υ22β(2)1υ22β(2)][Rδn0]=[M1cM1cM2cβ(1)M2cβ(1)][a2b2],
[c]mn=cnm,
[M1]mn=0Lbn(x)eiαmxdx,
[M2]mn=0Lbn(x)1υ12(x)eiαmxdx,
T([a1a2a3a4])=[a1a2a41a3a2a41a41a3a41].
[Rb2]=S2[a2δn0],
S2=T([II1υ22β(2)1υ22β(2)]1[M1cM1cM2cβ(1)M2cβ(1)]).
S21=[Iχ]S2[χI],
[Rb0]=S21[a0δn0],
[a0T]=S0[0b0],
S0=T([M1cM1cM2cβ(1)M2cβ(1)]1[II1υ02β(0)1υ02β(0)]1).
[RT]=S20[0δn0],
cos(γ1·pL)·cos(γ2·(1p)L)12(υ122γ1υ112γ2+υ112γ2υ122γ1)·sin(γ1·pL)·sin(γ2·(1p)L)=cos[2πLsin(θ)/λ],
(ψ,xp)=xpψ(x)dx=0,for0p<Q.
φj,p=a11φj,p(11)+a12φj,p(12)+a21φj,p(21)+a22φj,p(22).
Fa=0,
F=[φj,p(11)(pL)φj,p(12)(pL)φj,p(21)(pL+)φj,p(22)(pL+)1υ12(pL)dφj,p(11)dx(pL)1υ12(pL)dφj,p(12)dx(pL)1υ12(pL+)dφj,p(21)dx(pL+)1υ12(pL+)dφj,p(22)dx(pL+)],a=[a11a12a21a22]T.
(φj,p(m),φ˜j,p(n))=δmn,
[a1a2]=[X1X2][b1,1b1,2b2,1b2,2]=[X1X2]B,
BT[X1X2]T[X˜1X˜2]B˜=I2×2.

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